UNIVERSITY OF CINCINNATI Date: 4-Jun-2010 I, Kevin Henkener , hereby submit this original work as part of the requirements for the degree of: Doctor of Philosophy in Computer Science & Engineering It is entitled: Two-Hop f-Factors and a Fair and Trustworthy P2P Storage Model Student Signature: Kevin Henkener This work and its defense approved by: Committee Chair: Kenneth Berman, PhD Kenneth Berman, PhD Fred Annexstein, PhD Fred Annexstein, PhD Jerome Paul, PhD Jerome Paul, PhD Yiming Hu, PhD Yiming Hu, PhD Dan Ralescu, PhD Dan Ralescu, PhD 8/10/2010 1,011 Abstract In this dissertation, we present a novel approach to the problem of distributed (peer-to-peer) backup. Our approach requires that data not be transferred more than two-hops from it source and that each peer store exactly the same amount of data as it distributes to be backed up. These two requirements address two import features of any distributed backup solution - trust and fairness. In a social network, the hop distance requirement means that in the worst case, a peer’s data is backed up in the local storage of a friend of a friend (FoaF). Our assumption is that this offers a higher degree of trust than simply choosing a random peer. We achieve fairness through the requirement that peers store exactly the same amount of data that they distribute for backup. To facilitate this requirement, our approach uses symmetric exchanges of data. This not only supports fairness, but also enhances trust by introduc- ing a vested interest between peers to preserve the data that they are storing. We call our approach the fair two-hop exchange scheme, or FTHES. We show that existing f-factor theory and algorithms can be used to compute an FTHES. Then we introduce and prove a fundamental existence theorem which states that an FTHES always exists under two fairly weak conditions. This theorem leads to a linear time sequential algorithm and an efficient dis- tributed algorithm. We also prove a theorem stating that at most 2n − 3 exchanges are needed to backup all of the data in our scheme and later con- jecture that this may actually have a lower bound of n. Finally, we present an application of the FTHES in a content management system. This page is intentionally left blank. Acknowledgements I would first like to thank my parents, Randy and Sandy Henkener, for their continuous love and support throughout my life. They have always stood be- side me in my personal, professional, and academic careers. Without them, I would not be the person I am today. Second, I would like to thank my advisor, Dr. Kenneth Berman. He has served in several different roles in my life including instructor, advisor, and most importantly, friend. Dr. Berman was instrumental in obtaining various funding opportunities for me throughout my graduate school tenure. With- out his loyal support and dedication to my efforts, this work would not have been possible. I am indebted to him for his time and patience. Dr. Fred Annexstein, my co-advisor, was also a big part of this work. His patience, advice, support, and friendship have been invaluable. Thank you, Fred! I would also like to thank my other committee members, Dr. Jerome Paul, Dr. Yiming Hu, and Dr. Dan Ralescu, for their time and patience in serving on my committee. Last, but not least, I would like to thank my family and many friends that have supported me through the years. Thanks to Chad Yoshikawa and Svet- lana Strunjas for grinding through many years of classes and research along side me. Thanks to my college roommates, Scott Kovacs, Rick Burke, and Will Perry, for providing a welcome escape from the daily rigors of gradu- ate student life. Thanks to Mark Sole for opening doors that would have otherwise been locked, for opening my mind to other worlds of possibility, and most importantly, for being a loyal friend. Finally, thanks to the long list of other friends and family that cannot be mentioned here because these acknowledgements need to fit on one page. Table of Contents 1 Introduction 7 1.1 Our Model for Distributed P2P Backup ............. 9 1.2OurApproach-FairTwo-HopExchangeScheme....... 12 1.2.1 GraphTheoreticalModel................. 14 1.2.2 TheoremsandResults.................. 14 2 Background and Related Research 16 2.1P2P................................ 17 2.1.1 P2PFileSharing..................... 18 2.1.2 Backup/Archiving.................... 18 2.1.3 TrustandReputation................... 21 2.1.4 Fairness.......................... 21 2.1.5 Replication and Redundancy ............... 22 2.2GraphTheory........................... 24 2.2.1 f-Factors (Integer vs 0/1) ................. 24 2.2.2 k-Hop f-Factors...................... 26 1 2.3 Terminology ............................ 26 3 Fair Two-Hop Exchange Scheme (FTHES) 27 3.1 Notation and Definitions ..................... 28 3.2 Modeling Fair Two-Hop Exchange Scheme with Two-Hop f- factors............................... 29 3.3 Fundamental Existence Theorem ................ 31 3.4Near-FairExchangeScheme................... 40 4 Algorithm for Computing an FTHES 43 4.1 Outline ............................... 44 4.2 Auxiliary Functions ........................ 44 4.3ExchangesInvolvingStrictLocalMaximas........... 45 4.3.1 IdentifyStrictLocalMaximas.............. 45 4.3.2 Computing Exchanges .................. 45 4.4 Building an Ascending Forest .................. 48 4.5 Computing Remaining Exchanges ................ 48 4.5.1 Exchanges Involving Vertices at Depth ≥ 2....... 49 4.5.2 Exchanges Involving Vertices at Depth ≤ 2....... 52 4.6 Complexity Analysis . ...................... 53 4.6.1 SLMS........................... 53 4.6.2 Ascending Forest Operations ............... 54 4.6.3 Exchanges in Ascending Forest ............. 54 2 5 Distributed Algorithm for Computing an FTHES 55 5.1IdentifyStrictLocalMaximas.................. 56 5.2RemoveStrictLocalMaximasWithinTwoHops........ 56 5.3 Ascending Forest Generation ................... 58 5.4 Computing Exchanges ...................... 58 5.5 Complexity ............................ 61 5.5.1 MessagesInvolvingStrictLocalMaximas........ 61 5.5.2 Messages Involving Ascending Forest . ....... 62 5.5.3 Messages Involving Remaining Exchanges ....... 62 6 Minimizing FTHES Size 63 6.1FTHESSize............................ 63 6.2ExpectedNumberofStrictLocalMaximas........... 67 6.3 Minimum FTHES Size - NP-Hard ................ 73 7 MyBook: Application of FTHES 75 7.1CollaborativeGroups....................... 76 7.1.1 Structuring Collaborative Groups - MyBook (MBK) . 77 7.2 Compile, Organize, Visualize - MyBook Graphical User Interface 81 7.3ProposedP2PModel....................... 83 7.3.1 CollaborativeBook-CBK................ 83 7.3.2 PeerInterface....................... 84 7.3.3 Content Distribution - FTHES ............. 85 7.3.4 ContentTracking..................... 87 3 7.3.5 Monitoring........................ 88 8 Conclusions and Future Research 89 8.1 Future Research .......................... 90 8.2FTHES:ConsideringEdgeCapacities.............. 91 8.2.1 OverviewandNotation.................. 92 8.2.2 GenerateTreeCovers................... 92 8.2.3 Covering ......................... 93 8.2.4 Compute Exchanges in τ ................. 96 8.2.5 FutureWork........................ 96 4 List of Figures 1.1 An example P2P network with f(p) marked outside each vertex and a fair exchange scheme given by the equal size exchanges on the edges. .. 10 1.2 (a) Example star graph G; (b) Only one fair exchange is possible. .... 12 2.1 Example graph G (f-values marked outside each vertex) and corresponding f-factor φ(G). ............................ 25 3.1 From left-to-right. G, G = G with bivalent vertices inserted. Notice that in G we are forced to choose two-hop paths in order to find an f-factor - a two-hop f-factor. ......................... 30 3.2 G. .................................. 32 3.3 Strict local maximas (marked in double circles) within two hops of one another. Making the exchange ε(1, 8) = 2 reduces vertex 8 to a non-strict local maxima. ............................ 33 3.4 Ascending forest A built from G and f . ................ 36 3.5 The vertices in A having depth at least 2. ............... 37 3.6 The vertices in A with depth at most 2. ................ 38 5 6.1 The maximum number of strict local maximas (shown in red) in a graph G for which the neighborhood sufficiency condition holds is n − 2. .... 65 6.2 Pareto distributed f-values. The x-axis shows the different f-value sizes. The y-axis shows the number of peers having a given f-value. ...... 68 6.3 Randomly distributed f-values. The x-axis shows the different f-value sizes. The y-axis shows the number of peers having a given f-value. ... 69 6.4 Average number of strict local maximas for a graph containing 256 ver- tices with f-values assigned via a random distribution. The x-axis shows the number of edges. The y-axis shows the average number of strict local maximas for each edge count. ..................... 71 6.5 Average number of strict local maximas for a graph containing 256 ver- tices with f-values assigned via a Pareto distribution. The x-axis shows the number of edges. The y-axis shows the average number of strict local maximas for each edge count. ..................... 72 6.6 Reduction of the sum of subsets problem to the two-hop f-factor problem. 74 7.1 The general structure of an MBK. ................... 78 7.2 The MyBook GUI - left panel contains the topic hierarchy; top left panel contains a breadcrumb trail of topics in a given path; top right panel contains file references; bottom left panel is a built-in